Generalized Matrix Nearness Problems

We show that the global minimum solution of can be found in closed form with singular value decompositions and generalized singular value decompositions for a variety of constraints on involving rank, norm, symmetry, two-sided product, and prescribed eigenvalue. This extends the solution of Friedland–Torokhti for the generalized rank-constrained approximation problem to other constraints and provides an alternative solution for rank constraint in terms of singular value decompositions. For more complicated constraints on involving structures such as Toeplitz, Hankel, circulant, nonnegativity, stochasticity, positive semidefiniteness, prescribed eigenvector, etc., we prove that a simple iterative method is linearly and globally convergent to the global minimum solution.